A little more on ideals associated with sublocales | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 8، دوره 20، شماره 1، فروردین 2024، صفحه 175-200 اصل مقاله (589.79 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.48308/cgasa.2023.234093.1456 | ||
| نویسندگان | ||
| Oghenetega Ighedo* 1؛ Grace Wakesho Kivunga2؛ Dorca Nyamusi Stephen3 | ||
| 1Department of Mathematics, Chapman University, P.O. Box 92866, California, U.S.A. | ||
| 2Department of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003 Pretoria, South Africa. | ||
| 3Deparment of Mathematics and Physics, Technical University of Mombasa, P.O. Box 90420-80100, Mombasa, Kenya. | ||
| چکیده | ||
| As usual, let $\mathcal RL$ denote the ring of real-valued continuous functions on a completely regular frame $L$. Let $\beta L$ and $\lambda L$ denote the Stone-\v{C}ech compactification of $L$ and the Lindel\"of coreflection of $L$, respectively. There is a natural way of associating with each sublocale of $\beta L$ two ideals of $\mathcal RL$, motivated by a similar situation in $C(X)$. In~\cite{DS1}, the authors go one step further and associate with each sublocale of $\lambda L$ an ideal of $\mathcal RL$ in a manner similar to one of the ways one does it for sublocales of $\beta L$. The intent in this paper is to augment~\cite{DS1} by considering two other coreflections; namely, the realcompact and the paracompact coreflections.\\ We show that $\boldsymbol M$-ideals of $\mathcal RL$ indexed by sublocales of $\beta L$ are precisely the intersections of maximal ideals of $\mathcal RL$. An $\boldsymbol{M}$-ideal of $\mathcal RL$ is \emph{grounded} in case it is of the form $\boldsymbol{M}_S$ for some sublocale $S$ of $L$. A similar definition is given for an $\boldsymbol{O}$-ideal of $\mathcal RL$. We characterise $\boldsymbol M$-ideals of $\mathcal RL$ indexed by spatial sublocales of $\beta L$, and $\boldsymbol O$-ideals of $\mathcal RL$ indexed by closed sublocales of $\beta L$ in terms of grounded maximal ideals of $\mathcal RL$. | ||
| کلیدواژهها | ||
| Frame؛ locale؛ sublocale؛ pointfree function ring؛ Lindel\"{o}f؛ realcompact؛ paracompact | ||
| مراجع | ||
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